It follows simply by the pigeonhole principle. Here is an excerpt from my [old post][1]:
Recall (below) $\rm\; AB \:\:=\:\:\: I \:\;\Rightarrow\; BA \:\:=\:\: I\;\;\:$ easily reduces to:
THEOREM $\;$ $\rm\;\;B\;$ injective $\rm\;\Rightarrow\:\: B\;$ surjective, $\:$ for linear $\rm\:B\:$ on finite dim vector space $\rm\:V$
Proof: $\rm\;B\;$ injective $\rm\;\Rightarrow B\;$ preserves injections: $\rm\;R < S \;\Rightarrow\; BR < BS\;$
Hence for $\rm\;\;\; R \;\: < \;\; S < \cdots < \; V\;\;$ a chain of maximum length (= dim $\rm V\:$)
its image $\rm\;BR < BS < \cdots < BV \le V\;\;\:\;$ would have length greater
if $\rm\;BV < V,\:$ therefore instead $\rm\;\: BV = V\:,\;\:$ i.e. $\rm\; B \;$ is surjective. $\;$ QED
Notice how this form of proof dramatically emphasizes the essence of the matter, namely that
injective maps cannot decrease heights (here = dimension = length of max subspace chain).
Below is said standard reduction to xxxjective form. See said [sci.math post][1] for much more,
including references to folklore generalizations, e.g. work of Vasconcelos in the seventies.
First, notice that $\rm\;\;\: AB = I \;\Rightarrow\: B\:$ injective, since $\rm\;A\;$ times $\rm\;Bx = By\;$ yields $\rm\;x = y\:,\:$ and
$\rm B\:$ surjective $\rm\;\Rightarrow\; BA = I \;\;$ since for all $\rm\;x\;$ exists $\rm\; y : \;\; x = By = B(AB)y = BA \: x$
Combining them: $\rm\: AB = I \;\Rightarrow\: B\:$ injective $\rm\;\Rightarrow\; B\:$ surjective $\rm\;\Rightarrow\: BA = I$ [1]:http://google.com/[email protected]